Apart from working on the cellular decomposition \(\triangle \)  of the 3D spacetime manifold \({\mathcal {M}}\)  and the graph \(\Gamma =(\partial \triangle )^*_1\)  on the boundary of \(\triangle \) , we will also work on the dual cellular decomposition \(\triangle ^*\) .
In the dual picture, a dual face \(f^*\)  is dual to one edge \(e\) , an oriented dual edge \(e^*\)  is dual to one face \(f\) , and a dual vertex \(v^*\)  is dual to one 3-cell \(\sigma \) .
When \(e^*\)  is on the boundary of \(f^*\) , we denote \(e^*\in \partial f^*\) .
An example is illustrated in fig.REF , in which \(l_i\)  is dual to \(e_i\) , and the orientation of \(\Gamma \)  is chosen to be the Kasteleyn orientation, for the reason related to the duality between spin network and 2D Ising model [1]}. A Kasteleyn orientation is such that each plaquette has an odd number of links oriented opposite the orientation induced by the plaquette .

It was shown in [2]} that there exists a Kasteleyn orientation if and only if the number of nodes in a graph \(\Gamma \)  embedded in a surface \(\Sigma \)  is even, and there are \(2^{2g}\)  equivalence classes of Kasteleyn orientations of \(\Gamma \) , where \(g\)  is the genus of \(\Sigma \) . Two Kasteleyn orientations are called equivalent when one can be obtained from the other by a sequence of moves, in which the orientation of all the links incident to a node are reversed. Thus there is a unique equivalence class of Kasteleyn orientation for the graph \(\Gamma =(\partial T)^*_1\)  dual to the boundary of a tetrahedron.
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